Hexagonal tiling honeycomb

Hexagonal tiling honeycomb
File:H3 633 FC boundary.png Perspective projection view within Poincaré disk model
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbols	{6,3,3} t{3,6,3} 2t{6,3,6} 2t{6,3^[3]} t{3^[3,3]}
Coxeter diagrams	Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD ↔ Template:CDD ↔ Template:CDD ↔ Template:CDD ↔ Template:CDD
Cells	{6,3} Error creating thumbnail:
Faces	hexagon {6}
Edge figure	triangle {3}
Vertex figure	Error creating thumbnail: tetrahedron {3,3}
Dual	Order-6 tetrahedral honeycomb
Coxeter groups	${\overline{V}}_{3}$ , [3,3,6] ${\overline{Y}}_{3}$ , [3,6,3] ${\overline{Z}}_{3}$ , [6,3,6] ${\overline{V P}}_{3}$ , [6,3^[3]] ${\overline{P P}}_{3}$ , [3^[3,3]]
Properties	Regular

In the field of hyperbolic geometry, the hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity.

The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling is {6,3}, this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the tetrahedron is {3,3}, the vertex figure of this honeycomb is a tetrahedron. Thus, four hexagonal tilings meet at each vertex of this honeycomb, six hexagons meet at each vertex, and four edges meet at each vertex.^[1]

Images

Viewed in perspective outside of a Poincaré disk model, the image above shows one hexagonal tiling cell within the honeycomb, and its mid-radius horosphere (the horosphere incident with edge midpoints). In this projection, the hexagons grow infinitely small towards the infinite boundary, asymptoting towards a single ideal point. It can be seen as similar to the order-3 apeirogonal tiling, {∞,3} of H², with horocycles circumscribing vertices of apeirogonal faces.

{6,3,3}	{∞,3}
Error creating thumbnail:	File:Order-3 apeirogonal tiling one cell horocycle.png
One hexagonal tiling cell of the hexagonal tiling honeycomb	An order-3 apeirogonal tiling with a green apeirogon and its horocycle

Symmetry constructions

File:Hyperbolic subgroup tree 336-direct.png

Subgroup relations

It has a total of five reflectional constructions from five related Coxeter groups all with four mirrors and only the first being regular: Template:CDD [6,3,3], Template:CDD [3,6,3], Template:CDD [6,3,6], Template:CDD [6,3^[3]] and [3^[3,3]] Template:CDD, having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)^*] (remove 3 mirrors, index 24 subgroup); [3,6,3^*] or [3^*,6,3] (remove 2 mirrors, index 6 subgroup); [1⁺,6,3,6,1⁺] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3^[3,3]]. The ringed Coxeter diagrams are Template:CDD, Template:CDD, Template:CDD, Template:CDD and Template:CDD, representing different types (colors) of hexagonal tilings in the Wythoff construction.

Template:-

Related polytopes and honeycombs

The hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact. Template:Regular paracompact H3 honeycombs

It is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the order-6 tetrahedral honeycomb. Template:633 family

It is part of a sequence of regular polychora, which include the 5-cell {3,3,3}, tesseract {4,3,3}, and 120-cell {5,3,3} of Euclidean 4-space, along with other hyperbolic honeycombs containing tetrahedral vertex figures. Template:Tetrahedral vertex figure tessellations It is also part of a sequence of regular honeycombs of the form {6,3,p}, which are each composed of hexagonal tiling cells: Template:Hexagonal tiling cell tessellations

Rectified hexagonal tiling honeycomb

Rectified hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	r{6,3,3} or t₁{6,3,3}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD
Cells	{3,3} Error creating thumbnail: r{6,3} or
Faces	triangle {3} hexagon {6}
Vertex figure	Error creating thumbnail: triangular prism
Coxeter groups	${\overline{V}}_{3}$ , [3,3,6] ${\overline{P}}_{3}$ , [3,3^[3]]
Properties	Vertex-transitive, edge-transitive

The rectified hexagonal tiling honeycomb, t₁{6,3,3}, Template:CDD has tetrahedral and trihexagonal tiling facets, with a triangular prism vertex figure. The Template:CDD half-symmetry construction alternates two types of tetrahedra.

File:H3 633 boundary 0100.png

Hexagonal tiling honeycomb Template:CDD	Rectified hexagonal tiling honeycomb Template:CDD or Template:CDD
File:Hyperbolic 3d hexagonal tiling.png	File:Hyperbolic 3d rectified hexagonal tiling.png
Related H² tilings
Order-3 apeirogonal tiling Template:CDD	Triapeirogonal tiling Template:CDD or Template:CDD
File:H2-I-3-dual.svg	File:H2 tiling 23i-2.pngFile:H2 tiling 33i-3.png

Template:-

Truncated hexagonal tiling honeycomb

Truncated hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t{6,3,3} or t_0,1{6,3,3}
Coxeter diagram	Template:CDD
Cells	{3,3} t{6,3}
Faces	triangle {3} dodecagon {12}
Vertex figure	Error creating thumbnail: triangular pyramid
Coxeter groups	${\overline{V}}_{3}$ , [3,3,6]
Properties	Vertex-transitive

The truncated hexagonal tiling honeycomb, t_0,1{6,3,3}, Template:CDD has tetrahedral and truncated hexagonal tiling facets, with a triangular pyramid vertex figure.

Error creating thumbnail:

It is similar to the 2D hyperbolic truncated order-3 apeirogonal tiling, t{∞,3} with apeirogonal and triangle faces:

Template:-

Bitruncated hexagonal tiling honeycomb

Bitruncated hexagonal tiling honeycomb Bitruncated order-6 tetrahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	2t{6,3,3} or t_1,2{6,3,3}
Coxeter diagram	Template:CDD Template:CDD ↔ Template:CDD
Cells	t{3,3} t{3,6} File:Uniform tiling 63-t12.svg
Faces	triangle {3} hexagon {6}
Vertex figure	File:Bitruncated order-3 hexagonal tiling honeycomb verf.png digonal disphenoid
Coxeter groups	${\overline{V}}_{3}$ , [3,3,6] ${\overline{P}}_{3}$ , [3,3^[3]]
Properties	Vertex-transitive

The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t_1,2{6,3,3}, Template:CDD has truncated tetrahedron and hexagonal tiling cells, with a digonal disphenoid vertex figure.

File:H3 633-0110.png Template:-

Cantellated hexagonal tiling honeycomb

Cantellated hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	rr{6,3,3} or t_0,2{6,3,3}
Coxeter diagram	Template:CDD
Cells	r{3,3} File:Uniform polyhedron-33-t1.svg rr{6,3} File:Uniform tiling 63-t02.png {}×{3} File:Triangular prism.png
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	File:Cantellated order-3 hexagonal tiling honeycomb verf.png wedge
Coxeter groups	${\overline{V}}_{3}$ , [3,3,6]
Properties	Vertex-transitive

The cantellated hexagonal tiling honeycomb, t_0,2{6,3,3}, Template:CDD has octahedron, rhombitrihexagonal tiling, and triangular prism cells, with a wedge vertex figure.

File:H3 633-1010.png Template:-

Cantitruncated hexagonal tiling honeycomb

Cantitruncated hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	tr{6,3,3} or t_0,1,2{6,3,3}
Coxeter diagram	Template:CDD
Cells	t{3,3} tr{6,3} File:Uniform tiling 63-t012.svg {}×{3} File:Triangular prism.png
Faces	triangle {3} square {4} hexagon {6} dodecagon {12}
Vertex figure	File:Cantitruncated order-3 hexagonal tiling honeycomb verf.png mirrored sphenoid
Coxeter groups	${\overline{V}}_{3}$ , [3,3,6]
Properties	Vertex-transitive

The cantitruncated hexagonal tiling honeycomb, t_0,1,2{6,3,3}, Template:CDD has truncated tetrahedron, truncated trihexagonal tiling, and triangular prism cells, with a mirrored sphenoid vertex figure.

File:H3 633-1110.png Template:-

Runcinated hexagonal tiling honeycomb

Runcinated hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t_0,3{6,3,3}
Coxeter diagram	Template:CDD
Cells	{3,3} File:Uniform polyhedron-33-t0.png {6,3} Error creating thumbnail: {}×{6}File:Hexagonal prism.png {}×{3} File:Triangular prism.png
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	File:Runcinated order-3 hexagonal tiling honeycomb verf.png irregular triangular antiprism
Coxeter groups	${\overline{V}}_{3}$ , [3,3,6]
Properties	Vertex-transitive

The runcinated hexagonal tiling honeycomb, t_0,3{6,3,3}, Template:CDD has tetrahedron, hexagonal tiling, hexagonal prism, and triangular prism cells, with an irregular triangular antiprism vertex figure.

File:H3 633-1001.png Template:-

Runcitruncated hexagonal tiling honeycomb

Runcitruncated hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t_0,1,3{6,3,3}
Coxeter diagram	Template:CDD
Cells	rr{3,3} Error creating thumbnail: {}x{3} File:Triangular prism.png {}x{12} File:Dodecagonal prism.png t{6,3} Error creating thumbnail:
Faces	triangle {3} square {4} dodecagon {12}
Vertex figure	isosceles-trapezoidal pyramid
Coxeter groups	${\overline{V}}_{3}$ , [3,3,6]
Properties	Vertex-transitive

The runcitruncated hexagonal tiling honeycomb, t_0,1,3{6,3,3}, Template:CDD has cuboctahedron, triangular prism, dodecagonal prism, and truncated hexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.

Template:-

Runcicantellated hexagonal tiling honeycomb

Runcicantellated hexagonal tiling honeycomb runcitruncated order-6 tetrahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t_0,2,3{6,3,3}
Coxeter diagram	Template:CDD
Cells	t{3,3} File:Uniform polyhedron-33-t12.png {}x{6} File:Hexagonal prism.png rr{6,3} File:Uniform tiling 63-t02.png
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	Error creating thumbnail: isosceles-trapezoidal pyramid
Coxeter groups	${\overline{V}}_{3}$ , [3,3,6]
Properties	Vertex-transitive

The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t_0,2,3{6,3,3}, Template:CDD has truncated tetrahedron, hexagonal prism, and rhombitrihexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.

File:H3 633-1011.png Template:-

Omnitruncated hexagonal tiling honeycomb

Omnitruncated hexagonal tiling honeycomb Omnitruncated order-6 tetrahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t_0,1,2,3{6,3,3}
Coxeter diagram	Template:CDD
Cells	tr{3,3} File:Uniform polyhedron-33-t012.png {}x{6} File:Hexagonal prism.png {}x{12} File:Dodecagonal prism.png tr{6,3} File:Uniform tiling 63-t012.svg
Faces	square {4} hexagon {6} dodecagon {12}
Vertex figure	Error creating thumbnail: irregular tetrahedron
Coxeter groups	${\overline{V}}_{3}$ , [3,3,6]
Properties	Vertex-transitive

The omnitruncated hexagonal tiling honeycomb or omnitruncated order-6 tetrahedral honeycomb, t_0,1,2,3{6,3,3}, Template:CDD has truncated octahedron, hexagonal prism, dodecagonal prism, and truncated trihexagonal tiling cells, with an irregular tetrahedron vertex figure.

File:H3 633-1111.png Template:-

References

Template:Reflist

Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. Template:ISBN. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
The Beauty of Geometry: Twelve Essays (1999), Dover Publications, Template:LCCN, Template:ISBN (Chapter 10, Regular Honeycombs in Hyperbolic Space Template:Webarchive) Table III
Jeffrey R. Weeks The Shape of Space, 2nd edition Template:ISBN (Chapters 16–17: Geometries on Three-manifolds I,II)
N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups (1999), Volume 4, Issue 4, pp 329–353 [1] [2]
N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H³: p130. [3]

External links

John Baez, Visual Insight: {6,3,3} Honeycomb (2014/03/15)
John Baez, Visual Insight: {6,3,3} Honeycomb in Upper Half Space (2013/09/15)
John Baez, Visual Insight: Truncated {6,3,3} Honeycomb (2016/12/01)

↑ Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III

[1] Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III

[1]

Hexagonal tiling honeycomb

Contents

Images

Symmetry constructions